Factors, Multiples, Primes, Squares and Cubes

A factor divides the number with no remainder.

Division finds the missing factor.

Find factor pairs systematically, then put every factor in the list.

The multiples increase by nine each time.

List nearby multiples and check both limits.

A prime has exactly two positive factors.

1. The factors of 2 are 1 and 2. The factors of 17 are 1 and 17.
2. One has only one factor. Nine has factor 3, and fifteen has factors 3 and 5. The primes are 2 and 17.

A square uses two equal factors.

1. 8^2=8\times8=64.

Which number multiplied by itself gives 121?

1. 11\times11=121, so \sqrt{121}=11.

A cube uses three equal factors.

1. 4^3=4\times4\times4=16\times4=64.

Look for three equal factors, not 125 divided by three.

1. 5\times5\times5=125, so the cube root is 5.

List their factors and find the greatest shared value.

1. Factors of 18: 1,\ 2,\ 3,\ 6,\ 9,\ 18. Factors of 30: 1,\ 2,\ 3,\ 5,\ 6,\ 10,\ 15,\ 30.
2. Their common factors are 1,\ 2,\ 3,\ 6. The HCF is 6.

List positive multiples until the lists first meet.

1. Multiples of 6: 6,\ 12,\ 18,\ 24,\ 30,\ 36,\ 42,\ldots. Multiples of 14: 14,\ 28,\ 42,\ldots.
2. The first shared positive value is 42.

Check the squares of nearby whole numbers.

1. 8^2=64, 9^2=81 and 10^2=100.
2. Only 81 is greater than 80 and strictly less than 100.

Find the multiples in the range, then test whether they divide 48.

1. The multiples of 6 in the range are 12 and 18.
2. 48\div12=4 exactly, but 18 does not divide 48 exactly. The number is 12.

Check whether each number can be made from two equal factors and from three equal factors.

1. 64=8\times8=8^2 and 64=4\times4\times4=4^3.
2. The other listed numbers are squares but not whole-number cubes. The required number is 64.

The number of bags must divide both colour totals.

Part (a)

1. Common factors of 36 and 48 are 1,\ 2,\ 3,\ 4,\ 6,\ 12.
2. The greatest is 12, so there can be at most 12 identical bags.

Part (b)

1. Each bag has 36\div12=3 red pens and 48\div12=4 blue pens.
2. Total per bag: 3+4=7 pens.

Find their repeating common interval before counting coincidences.

Part (a)

1. Multiples of 12: 12,\ 24,\ 36,\ldots. Multiples of 18: 18,\ 36,\ldots.
2. They next signal together after 36 minutes.

Part (b)

1. The common signals are at 36,\ 72,\ 108,\ 144 and 180 minutes.
2. There are 5 times. The starting signal is excluded, but the one at 180 minutes is included.

List the square numbers between 40 and 70 first.

1. 6^2=36, 7^2=49, 8^2=64 and 9^2=81, so the square numbers between 40 and 70 are 49 and 64.
2. 49\div7=7 exactly, but 64\div7 is not a whole number.
3. Therefore the number is 49.

Remove the three leftovers. What must be true of the remaining number?

1. After setting aside 3 chairs, the number left is a common multiple of 8 and 12.
2. Their LCM is 24, so possible totals are 24+3=27, 48+3=51, 72+3=75, 96+3=99, and so on.
3. Only 75 is greater than 60 and less than 90. Check: 75=9\times8+3 and 75=6\times12+3.

Find how many cubes are along an edge, then how many are in one layer.

1. 6^3=216, so there are 6 cubes along each edge.
2. Each layer has 6\times6=36 cubes. Two layers contain 2\times36=72 cubes.
3. The number remaining is 216-72=144.